| L(s) = 1 | + 2.68·2-s + 3-s + 5.19·4-s + 2.68·6-s + 3.41·7-s + 8.57·8-s + 9-s + 0.785·11-s + 5.19·12-s + 0.625·13-s + 9.17·14-s + 12.6·16-s − 1.30·17-s + 2.68·18-s − 7.31·19-s + 3.41·21-s + 2.10·22-s − 4.48·23-s + 8.57·24-s + 1.67·26-s + 27-s + 17.7·28-s + 0.982·29-s − 1.96·31-s + 16.6·32-s + 0.785·33-s − 3.49·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.59·4-s + 1.09·6-s + 1.29·7-s + 3.03·8-s + 0.333·9-s + 0.236·11-s + 1.50·12-s + 0.173·13-s + 2.45·14-s + 3.15·16-s − 0.315·17-s + 0.632·18-s − 1.67·19-s + 0.745·21-s + 0.449·22-s − 0.935·23-s + 1.75·24-s + 0.328·26-s + 0.192·27-s + 3.35·28-s + 0.182·29-s − 0.352·31-s + 2.94·32-s + 0.136·33-s − 0.598·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.093290744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.093290744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 0.785T + 11T^{2} \) |
| 13 | \( 1 - 0.625T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 - 0.982T + 29T^{2} \) |
| 31 | \( 1 + 1.96T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 - 4.01T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 + 0.112T + 61T^{2} \) |
| 67 | \( 1 - 9.41T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 + 9.51T + 73T^{2} \) |
| 79 | \( 1 + 8.50T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.200569762539803916271582052210, −7.80161381595495146144191453122, −6.77293039865738046102614181388, −6.25169404265797400914288667136, −5.32185403102503240613721569774, −4.56436150431923687608113036372, −4.12459721871550716252988862744, −3.25560240357172134127908838714, −2.15306394191690340093790851674, −1.71571504787310880373659206835,
1.71571504787310880373659206835, 2.15306394191690340093790851674, 3.25560240357172134127908838714, 4.12459721871550716252988862744, 4.56436150431923687608113036372, 5.32185403102503240613721569774, 6.25169404265797400914288667136, 6.77293039865738046102614181388, 7.80161381595495146144191453122, 8.200569762539803916271582052210
